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Register a new userRokhlin dimension: absorption of model actions
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Authors
Gabor Szabo
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In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing C*-algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $mathcal D$ is a strongly self-absorbing C*-algebra, and $alpha: Gcurvearrowright A$ is an action on a separable, $mathcal D$-absorbing C*-algebra that has finite Rokhlin dimension with commuting towers, then $alpha$ tensorially absorbs every semi-strongly self-absorbing $G$-actions on $mathcal D$. This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some non-trivial applications. Most notably it is shown that for any $kgeq 1$ and on any strongly self-absorbing Kirchberg algebra, there exists a unique $mathbb R^k$-action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.
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Let $(X, Gamma)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $Gamma$ is a discrete amenable group. It is shown that, if $(X, Gamma)$ has a version of Rokhlin property (uniform Rokhlin property) and if $mathrm{C}(X)rtimesGamma$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $mathrm{C}(X) rtimes Gamma$ is at most half of the mean topological dimension of $(X, Gamma)$. These two conditions are shown to be satisfied if $Gamma = mathbb Z$ or if $(X, Gamma)$ is an extension of a free Cantor system and $Gamma$ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.
We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimotos conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, strongly purely infinite C*-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable $KK$-contractible C*-algebras: Two Rokhlin flows on such a C*-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.
Consider a minimal free topological dynamical system $(X, T, mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, mathbb{Z}^d)$. As a consequence, the C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is classifiable if $(X, T, mathbb{Z}^d)$ has zero mean dimension.
We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $mathcal{O}_infty$ and $mathcal{O}_2$. This generalizes results known for finite groups and poly-$mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a precursor to a general Kirchberg-Phillips-type classification theory: If two outer actions of an amenable group on a unital Kirchberg algebra are equivariantly homotopy equivalent, then they are conjugate. This marks the first C*-dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups.
Morita equivalence of twisted inverse semigroup actions and discrete twisted partial actions are introduced. Morita equivalent actions have Morita equivalent crossed products.
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